Samples
and
populations

Probability
theory and statistical inferrence

Inferring a population
mean

Tests of
statistical significance

Tests of
differences between
means

Coefficient
of
correlation

Caution
with
association

Chi square

Other
tests of
significance

Caution in
using
statistical
test results

Aids

Aids

Key terms

Main points

1. Analysis of data from samples produces results called statistics. Examples of statistics include means, percentages, and standard deviations. Corresponding values in a population referred to as parameters.
2. Values for statistics vary from one sample to the next because of variations resulting from random selection of samples.
3. Inferential statistics are used to estimate parameters from statistic.
4. Inferential analysis can be used to estimate a single parameter, such as population mean from a sample mean, or to establish a relationship between two variables.
5. The exact value of a parameter, such as a population mean, cannot be estimated precisely. Instead, we can only estimate an interval in which the mean is probably located. This interval is known as the confidence interval and is defined in terms of standard errors above and below the sample mean. Further, the confidence interval is defined in terms of levels of confidence. The levels of confidence express how confident we want to be that the result was not due to chance variation. The usual confidence levels are estimating that a parameter is within the specified confidence interval is 95 times out of 100, referred to as the .05 level of confidence or that the parameter will be in the specified interval 99 times out of 100, referred to as the .01 level of confidence.
6. Tests of statistical significance are used in deciding whether a relationship between two variables observed in a sample also exist between the variables in the population from which the sample was selected.
7. Statistical tests of significance are based on testing the null hypothesis. This hypothesis states that two variables are not associated or that two statistics, such as means, are not different. Tests of significance establish whether the observed result is one that could be expected due to chance variations in sampling. When the probability is high that an association or difference could be due to sampling variation, the null hypothesis is accepted. When the probably is low that an association or difference could have occurred due to chance, the null hypothesis is rejected.
8. Traditionally, the .05 level of significance is used in testing the null hypothesis. This level says that a result could have occurred due to chance in less than 5 out of every 100 samples that could be selected from a population. With a result significant at the .05 level, we could be wrong in rejecting the null hypothesis 5 times in 100. At the .01 level of significance, we would incorrectly reject the null hypothesis 1 time in 100.
9. When the null hypothesis is accepted, the conclusion is that no relationship exists between the variables being analyzed. When the null hypothesis is rejected, the alternative hypothesis established at the beginning of the investigation, also called the research hypothesis, can be accepted.
10. The t test is used to test the significance of the difference between two means.
11. Pearson's coefficient of correlation is used for measuring the association between two continuous variables (measured at the interval or ratio levels). It varies between -1.0 and +1.0. A scatter plot, based on plotting pairs of observations on the X and Y coordinates of a graph, can show the direction and degree of association between variables. The independent variable is traditionally labeled as the X variable and the dependent variable as Y.
12. Spearman's rank order coefficient of correlation is used to test the association between pairs of ordinal, interval, or ratio scores that have been converted to ranks. The ranks are then used in place of the original scores in testing for association.
13. Chi square is used to test for the dependence between nominal or ordinal variables.
14. Statistical tests of significance are interpreted in terms of degrees of freedom, which depend on the N used in the analysis. Critical values for tests, such as the t test or chi square, indicate the levels of significance at various levels of confidence (.05, .01, or .001). When the result for a test exceeds the specified critical value at the specified degrees of freedom, the null hypothesis is rejected. When the result is less than the critical value at the specified degrees of freedom, the null hypothesis is accepted.
15. Statistical tests do not mean that results have theoretical value or practical importance. With a large enough N, almost any result can be statistically significant. In addition to assessing the statistical significance of results, researchers also have to make judgments about the theoretical or practical value of findings.
16. Strictly speaking, tests of significance should only be used to analyze data from properly drawn probability samples. Nevertheless, tests of significance are used with nonprobability samples. These tests are useful for establishing the extent of relationships among variables, even though the conclusions cannot be safely generalized to any population.